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Moore Graphs and Beyond: Recent Advances in the Degree/Diameter Problem Mirka Miller King’s College London, UK & University of Newcastle, Australia & University of West Bohemia, Czech Republic & ITB Bandung, Indonesia mirka.miller@gmail.com UPC Barcelona, October 2012 Priority Research Centre for Computer-Assisted Research Mathematics and its Applications Graph Theory and Applications research group
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2 Outline of the talk Degree/diameter problem Some techniques Some open problems Prove non-existence of almost Moore digraphs Construct some missing mixed Moore graphs or prove their non- existence Construct radial Moore graphs or prove their non-existence
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3 Open Problems 1. Does there exist a regular directed graph with degree d>3, diameter k>4 and n=d +d 2 + … + d k vertices? 2. Does there exist a regular mixed graph with undirected degree 3, out-degree 3, diameter 2 and 40 vertices? 3. Find a general construction for undirected graphs of degree 3, order n= (3 k+1 – 1)/2, and diameter at most k+1. k=0, n=1 k=1, n=4 k=2, n=13 k=3, n=40 k=4, n=121 k=5, n=364 etc.
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4 Degree/diameter problem Degree/diameter problem (proposed by Elspas, 1964): Determine the largest number of vertices N(d,k) of a graph G for given maximum degree d and diameter at most k. Directed version: Determine the largest number of vertices N(d,k) of a digraph G for given maximum out-degree d and diameter at most k. Mixed version: Determine the largest number of vertices N(r,z,k) of a mixed graph G for given maximum undirected degree r, maximum out-degree z (d=r+z) and diameter at most k.
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6 Almost Moore Digraphs 1. Do there exist directed graphs with maximum out- degree d, diameter k and the number of vertices one less than the Moore bound M d,k ?
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7 Moore bound A natural upper bound on the number of vertices n of digraph G with given maximum out-degree d and diameter at most k is: n = 1+d +d 2 + … + d k. This bound M d,k is called the Moore bound. A digraph attaining this bound is called a Moore digraph M(d,k).
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8 Moore digraphs Examples of Moore digraphs M(d.k) – an undirected edge represents two arcs of opposite directions C3C3 C5C5 Plesnik & Znam, 1974, Bridges & Toueg, 1980 : Moore digraphs exist only in trivial cases, for d =1 (directed cycles of k+1 vertices) or k =1(complete digraphs on d+1 vertices).
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9 Almost Moore digraphs Since Moore digraphs do not exist for all d >1 and k >1, we are interested in digraphs that are “close to being Moore”. Relax order to M d,k – (defect 1 If = 1 almost Moore digraphs or digraphs of defect 1. Notation. (d,k)-digraph is a digraph of maximum out-degree d, diameter k and n = M d,k –1 vertices. 1 2 56 3 4 1 2 3 4 5 6 1 2 3 4 5 6
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10 Almost Moore digraphs Fiol,Alegre,Yebra, ‘83: (d,2)-digraphs exist for any d 2. Example: The line digraph L(K d+1 ) of the complete digraph K d+1 (Kautz digraphs). 56 3 4 1 2 6 2 3 1 5 4 MM,Fris, 1992: There are no (2,k)-digraphs for any k 3. Baskoro,MM,Siran,Sutton, 2005: There are no (3,k)- digraphs for any k 3.
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11 Almost Moore digraphs Gimbert in 2001 showed that line digraphs are the only (d,2)-digraphs for d 3. Conde,Gimbert,Gonzalez,Miret,Moreno, 2008: There are no (d,3)-digraphs for any d 3. Do there exist any (d,k)-digraphs, d 4, k 4? Lleida Mafia : NO for k=4. So next smallest unknown case is n=4+16+64+256+1024=1364, d=4, k=5 Now: Do there exist any (d,k)-digraphs, d 4, k 5?
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12 Repeats Could be useful for Problems 1 and 2.
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13 Almost Moore digraphs: repeats Let G be a (d,k)-digraph. ( MM&Fris, 1988 ) For every vertex x of G there exists a vertex y, called the repeat of x, such that there are two walks of lengths k from x to y. Write y = r(x). N+(S): the multiset of all out-neighbours of the vertices in set S. r(S): the multiset of all the repeats of the vertices in S.
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14 Neighbourhood Theorem Neighbourhood Theorem. (Baskoro, MM, Plesnik, Znam ’95) Let G be a diregular (d,k)-digraph. Then N + (r(u)) = r(N + (u)). The mapping r is an automorphism, permutation
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15 Neighbourhood Theorem Proof. By listing in two different ways the multiset of vertices that can be reached from a vertex v in at most k+1 steps.
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16 AMDs: regularity Almost Moore digraphs for any d and k>2 are out-regular – by simple counting argument in-regular – not so simple MM, Gimbert, Siran, Slamin, ’00. For k>2, all digraphs of maximum out-degree d>1 and order 1 less than the Moore bound are diregular. So we can use Neighbourhood Theorem for all almost Moore digraphs.
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17 Almost Moore digraphs Repeat order of a vertex = (v) is the smallest number such that r (v) = v. Further study may focus on the existence of (d,k)-digraphs with selfrepeats (d,k)-digraphs without selfrepeats
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18 Almost Moore digraphs with selfrepeats Theorem. For d>1, k>2, every (d,k)-digraph contains either k selfrepeats or none. [Baskoro, MM & Plesnik, 1998] In other words, there is one C k or none. Note that the girth (length of shortest cycle) of AMD is always k or k+1. Useful lemmas: 1.For any selfrepeat v in a (d,k)-digraph G, the permutation r on N+(v) has the same structure as the permutation r on N (v). 2.The permutation r on N+(v) has the same structure for every selfrepeat v in a (d,k)-digraph.
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19 Almost Moore digraphs with selfrepeats OUTLINE OF THE PROOF. Let V 1 be the set of all selfrepeats in G. Consider the induced subdigraph G[V 1 ]. Since in G any walk of length k joining any two selfrepeat vertices involves only selfrepeats then the diameter G[V 1 ] is at most k. If all vertices of G[V 1 ] have outdegree 1 then there are exactly k vertices in G[V 1 ]; and G[V 1 ] is a k-cycle. If there exists a vertex in G[V 1 ] with outdegree d 1 2 then G[V 1 ] is diregular, degree d 1 and order d 1 +d 1 2 +…+d 1 k almost Moore digraph! BUT…
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20 Almost Moore digraphs with selfrepeats OUTLINE OF THE PROOF that there are no (d 1,k)- digraphs with all vertices selfrepeats, d 1 >1, k>2. I+A+…+A k =J+I, where A is the adjacency matrix of G, J is unit matrix, I is identity matrix A+…+A k =J J has eigenvalues n (once) and 0 (n-1 times) A has eigenvalues d (once) and n-1 roots of the characteristic equation x+x 2 +… +x k = 0 x = 0 and roots of x k -1 =0 Since tr(A j )=0 for 0 j k-1, we obtain –d = –d k-1 and so d =1 or k =2 are the only solutions.
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21 Almost Moore digraphs d 4, k 5 (d,k)-digraphs with selfrepeats There are k selfrepeats Given the repeats structure of the out-neighbourhood of one selfrepeat vertex, we can determine the repeats structure of the digraph… this could be helpful… (d,k)-digraphs without selfrepeats Totally open Current directions to prove non-existence: Lleida using algebraic methods; Pilsen divisibility conditions based on the lengths of small cycles; Malang based on vertex types and counting in two different ways – none finished.
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22 Mixed Moore Graphs 2. Does there exist a regular mixed graph with degree 3, out-degree 3, diameter 2 and 40 vertices?
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23 Moore bound - directed Let G be a digraph with given out-degree d and (a) diameter at most k (b) girth at least k+1. What is the (a) maximum (b) minimum possible number of vertices in G? n M* d,k = 1+d +d 2 + … + d k v This bound is called the Moore bound (for digraphs). A digraph attaining this bound is called a Moore digraph. Plesnik & Znam, ’74, Bridges & Toueg, ’80 : Moore digraphs exist only for trivial cases, for d =1 (directed cycles of k+1 vertices) or k =1 (complete digraphs on d+1 vertices). No open problems.
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24 Moore bound Let G be a graph of degree d and (a) diameter at most k (b) girth at least 2k + 1. What is the (a) maximum (b) minimum number of vertices in G? n M d,k =1+d+d(d-1)+…+d(d-1) k-1 This bound is called the Moore bound.. Graph attaining this bound is called a Moore graph. v
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25 Moore graphs k =1: Moore graphs are complete graphs K d+1. k =2: Moore graphs exist for d =2 (cycle C 5 ) or d =3 (Petersen graph) or d =7 (Hoffman-Singleton graph) or d =57(?) k =3: Moore graph exists for d =2 (cycle C 7 ). Hoffman&Singleton,’60 k 3: Moore graphs are C 2k+1. Damerell, ‘73; Bannai & Ito, ’73 One open problem: k=2, d=57.
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26 A mixed graph may contain (undirected) edges as well as (directed) arcs. A proper mixed graph has at least 1 edge and 1 arc. Mixed graphs v E3E3 E2E2 E4E4 E1E1 E5E5
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27 Moore bound for mixed graphs v If r=0 then d=z and get directed Moore bound If z=0 d=r undirected Moore bound Theorem. There are no mixed Moore graphs of diameter > 2. Nguyen, Gimbert, MM, ‘07
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28 Mixed Moore graphs of diameter 2 Bosak graph: Proper mixed Moore graph of order 18, z=1, r=3. Kautz digraph: Proper mixed Moore graph of order 12, z=2, r=1.
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29 Mixed Moore graphs of diameter 2 (Bosák,1979) A mixed Moore graph of order n, degree d and diameter k=2 has the following properties: i. Totally regular, undirected degree r and directed degree z (d=z+r) ii. iii. either or or Many open problems: k=2, many d.
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30 Mixed Moore graph s Mixed Moore graph with n=40, d=6, z=3, r=3, k=2 can be considered as a regular digraph of diameter 2, degree 6, and M 6,2 – 3 vertices, with every vertex having 3 selfrepeats. Recent new result: Jorgensen constructed mixed Moore graph of 108 vertices with undirected degree 3, directed degree 7 and diameter 2.
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31 Mixed Moore graphs of diameter 2 ndzrExampleUniqueness (up-to-isomorphism) 3110Yes 5202C5C5 6211Ka(2)Yes 10303Petersen graphYes 12321Ka(3)Yes 18413Bosak graphYes 20431Ka(4)Yes 30541Ka(5)Yes 40633?? 42651Ka(6)Yes 50707Hoffman-Singleton graphYes 54743?? 56761Ka(7)Yes 72871Ka(8)Yes 84927?? 88963?? 90981Ka(9)Yes
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32 Mixed Moore graphs of diameter 2 (cont.) ndzrExampleUniqueness (up-to-isomorphism) 1041037?? 1061055?? 1081073Jorgensen graph? 1101091Ka(10)Yes 1261147?? 1281165?? 1301183?? 13211101Ka(11)Yes 1501257?? 1521275?? 1541293?? 15612111Ka(12)Yes 1761367?? 1781385?? 18013103?? 18213121Ka(13)Yes 19814113??
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33 Mixed Moore graphs of diameter 2 (cont.) ndzrExampleUniqueness (up-to-isomorphism) 2041477?? 2061495?? 20814113?? 21014131Ka(14)Yes 22815213?? 2341587?? 23615105?? 23815123?? 24015141Ka(15)Yes 26016313?? 2661697?? 26816115?? 27016133?? 27216151Ka(16)Yes 29417413?? 30017107??
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34 Radial Moore Graphs 3. Find a general construction for undirected graphs of degree 3, order n= (3 k+1 – 1)/2, and diameter k+1.
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35 Radial Moore graphs and digraphs Digraph G is a radially Moore digraph if it has maximum out-degree d, radius k, order M d,k and the diameter of G does not exceed k+1. (Knor,1996) Radially Moore digraph of degree d and radius k exists for every d and k. What about (undirected) radial Moore graphs? G is a radial Moore graph if it has degree d, radius k, order M d,k and the diameter of G does not exceed k+1.
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36 Radial Moore graphs (Knor,1996) Radial Moore digraph of degree d and radius k exists for every d and k. What about radial Moore graphs? G is a radial Moore graph if it has degree d, radius k, order M d,k = 1+d+d(d-1)+…+d(d-1) k-1 and the diameter of G does not exceed k+1. (Exoo, Gimbert, Lopez, Gomez, 2012) Radial Moore graph of degree d and radius k=3 exists for every d. (with some input from Knor) Current general conjecture due to Exoo: Apart from the known radial Moore graphs no others exist. Anti-Exoo conjecture: There are more radial Moore graphs.
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37 Radial Moore graphs All cubic radial Moore graphs of radius 2.
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38 Radial Moore graphs What about cubic radial Moore graphs of radius greater than 2? Cases k=3, 4, 5 are known (Exoo, Baladram) What about k>5? Do they all exist? Why do we care?
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39 Current table of largest known graphs
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40 Radial Moore graphs Why do we care about cubic radial Moore graphs? We would get infinitely many new largest graphs current largestradial Moore k=8 336 382 k=9 600 766 k=1012501534 …
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41 http://www.mat.upc.es/grup_de_grafs/grafs/taula_delta_d.html http://combinatoricswiki.org/wiki/Main_Page Charles Delorme by A. Brzozowski Francesc ComellasEyal Loz, Hebert Perez-Roses, Guillermo Pineda-Villavicencio
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42 ADVERTISEMENT http://combinatoricswiki.org/wiki/Main_Page Eyal Loz Hebert Perez-Roses Guillermo Pineda- Villavicencio
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43 Combinatorics Wiki: A wiki presenting the latest concepts, results, conjectures and references in various topics of Combinatorics. Combinatorics Wiki is regularly updated by a global community of combinatorialists. Currently extending the list of users, research topics, lectures and documentaries, and so new contributors are welcome (just send an email to eyal@math.auckland.ac.nz for an account and some wiki help). Combinatorics Wiki (established by Eyal Loz, Hebert Perez- Roses & Guillermo Pineda-Villavicencio)
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44 http://combinatoricswiki.org/wiki/Main_Page Combinatorics Wiki
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45 http://combinatoricswiki.org/wiki/Main_Page Combinatorics Wiki
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46 Thank you
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